Question

Let A,B, C be three events suich that P(A)=0.4, P(C)=0.3, P((A ) ̅ ∩B)=0.2 and P(A∩B)=0.1 and
P(A∩B)=0.1 and (A∪B)∩C=∅
Find a) P(A∪B∪C) b) P(A ̅ ∪ B ̅)

Answer 1

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Answer on Question #69191 – Math – Statistics and Probability

Question

Let $A, B, C$ be three events such that $P(A) = 0.4$, $P(C) = 0.3$, $P((A \succ \cap B) = 0.2$ and $P(A \cap B) = 0.1$ and $(A \cup B) \cap C = \emptyset$.

Find:

a) $P(A \cup B \cup C)$

b) $P(A \cup B)$

Solution

a) Since, $(A \cup B) \cap C = \emptyset$,

$$\begin{array}{l} P(A \cup B \cup C) = P(A \cup B) + P(C) = P(A) + P(B) - P(A \cap B) + P(C) \\ = P(A) + P\left((A \cap B) \cup (\overline{A} \cap B)\right) - P(A \cap B) + P(C) \\ = P(A) + P(A \cap B) + P(\overline{A} \cap B) - P(A \cap B \cap \overline{A} \cap B) - P(A \cap B) \\ + P(C) = P(A) + P(\overline{A} \cap B) - P(\emptyset) + P(C) \\ = P(A) + P(\overline{A} \cap B) + P(C) = \\ = 0.4 + 0.2 + 0.3 = 0.9, \end{array}$$

so $P(A \cup B \cup C) = 0.9$.

b)

$$P(\overline{A} \cup \overline{B}) = P(\overline{A \cap B}) = 1 - P(A \cap B) = 1 - 0.1 = 0.9,$$

So $P(\overline{A} \cup \overline{B}) = 0.9$.

Answer: a) $P(A \cup B \cup C) = 0.9$; b) $P(\overline{A} \cup \overline{B}) = 0.9$.